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Chapter 8: Problem 363

Why is there no solution to the equation \(\frac{3}{x-2}=\frac{5}{x-2} ?\)

Short Answer

Expert verified
The equation \(\frac{3}{x-2}=\frac{5}{x-2}\) has no solution because \(3\) is not equal to \(5\).

Step by step solution

01

Identifying the Equation

The equation given is \(\frac{3}{x-2}=\frac{5}{x-2}\).
02

Assuming Cross Multiplication

To solve, notice that since the denominators are already the same, we can equate the numerators: \(3 = 5\).
03

Analyzing the Result

Since \(3 = 5\) is a false statement, it indicates that there is no value for \(x\) that can make the original equation true.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

rational equations
Rational equations involve fractions with polynomials in the numerator, denominator, or both. The key feature of these equations is the presence of variables in the denominator. They require special techniques to solve, often involving making the denominators the same or using cross multiplication. For this problem, consider the rational equation \(\frac{3}{x-2}=\frac{5}{x-2}\). Since both sides share the same denominator, we can simplify the approach.

We equate the numerators directly to make it easier.
cross multiplication
Cross multiplication is a common method used to solve rational equations. It helps to eliminate the fractions by multiplying both sides of the equation by the product of the denominators. This method is particularly useful when the denominators are different.

However, in the problem \(\frac{3}{x-2}=\frac{5}{x-2}\), the denominators are already the same. Thus, instead of cross multiplication, we simply equate the numerators:
\[ 3 = 5 \]
This step eliminates the variables and allows us to compare the remaining numbers directly.
false statements in algebra
When you solve an equation and end up with a false statement, like \``3=5\,'' it tells us that there is no solution. This means there's no possible value for the variable that will satisfy the original equation.

In the context of the given problem, the equation \(\frac{3}{x-2}=\frac{5}{x-2}\) leads to a contradiction when simplified.
Therefore, we conclude that there is no value for \x\ that can make the equation true because \``3=5\'' is always false.
  • No solution arises from the fact that the equation simplifies to a false statement.
  • Always look out for these contradictions as they are crucial in algebra.

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